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CSTR: Multiplicidade de Soluções e Análise de Estabilidade
Argimiro R. SecchiPrograma de Engenharia Química – COPPE/UFRJ
Rio de Janeiro, RJ
EQE038 – Simulação e Otimização de Processos Químicos
EQ/UFRJmarço de 2014
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-Multiplicity of steady states
-Linearization
-System stability
-Complex dynamic behaviors (limit cycles, strange attractors)
-Parametric sensitivity and input sensitivity
System AnalysisSystem Analysis
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Non-isothermal CSTR
Fe , CAf , CBf , Tf
Fwe , Twe
Fws , Tw
Fs , CA , CB , T
V , T
A k B
Multiplicity of Steady StatesMultiplicity of Steady States
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In a non-isothermal continuous stirred tank reactor, with diameter of 3.2 m
and level control, pure reactant is fed at 300 K and 3.5 m3/h with concentration
of 300 kmol/m3. A first order reaction occur in the reactor, with frequency factor
of 89 s-1 and activation energy of 6 x 104 kJ/kmol, releasing 7000 kJ/kmol of
reaction heat. The reactor has a jacket to control the reactor temperature, with
constant overall heat transfer coefficient of 300 kJ/(h.m2.K). Assume constant
density of 1000 kg/m3 and constant specific heat of 4 kJ/(kg.K) in the reaction
medium. The fully-open output linear valve has a constant of 2.7 m2.5/h.
Process description
Multiplicity of Steady StatesMultiplicity of Steady States
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•perfect mixture in the reactor and jacket;
•negligible shaft work;
•(-rA) = k CA;
•constant density;
•constant overall heat transfer coefficient;
•constant specific heat;
•incompressible fluids;
•negligible heat loss to surroundings;
(internal energy) (enthalpy);
•negligible variation of potential and kinetic energies;
•constant volume in the jacket;
•thin metallic wall with negligible heat capacity.
Model assumptions
Multiplicity of Steady StatesMultiplicity of Steady States
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dt
dVFF
dt
Vdsef
)(
se FFdt
dV
)( AAsfAeA
AA rVCFCFdt
dVC
dt
dCV
dt
VCd
Mass balance in the reactorOverall:
Component:
(2)VrCCFdt
dCV AAfAe
A )()(
(1)
eF
V (3)
CSTR modeling
Multiplicity of Steady StatesMultiplicity of Steady States
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(4)
2 2
ˆˆ ˆ ˆ ˆ ˆ ˆ2 2f s
e f f f f s s r s
v vdV U K F U P V gz F U PV gz q q w
dt
ˆ ˆ ˆH U PV
ˆ ˆ( ) ˆ ˆ ˆe f s r
d VH dH dVV H F H F H q q
dt dt dt
Energy balance in the reactor:
where
ˆˆ ˆ( )e f r
dHV F H H q q
dt
qqTTCpFdt
dTVCp rfe )(
CSTR modeling
Multiplicity of Steady StatesMultiplicity of Steady States
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(5)where
qr = (-Hr) V (-rA)
k = k0 exp(–E/RT)
(-rA) = k CA
V = A h
Fs = x Cv h
Tw = f(T)
(7)
(6)
(8)
(10)
(12)
Temperature control (14)
At = A + D h (11)
q = U At (T – Tw)
A = D2/4 (9)
x = f(h) Level control (13)
CSTR modeling
Multiplicity of Steady StatesMultiplicity of Steady States
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variable units of measurement
Fe, Fs m3 s-1
V m3
t, s
CA, CAf kmol m-3
rA kmol m-3 s-1
kg m-3
Cp kJ kg-1 K-1
T, Tf, Tw K
qr, q kJ s-1
U kJ m-2 K-1 s-1
At, A m2
h, D m
Cv m2.5 h-1
x –
Hr, E kJ kmol-1
R kJ kmol-1 K-1
k, k0 s-1
Consistency analysis
Multiplicity of Steady StatesMultiplicity of Steady States
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variables: Fe, Fs, V, t, CA, CAf, rA, , Cp, T, Tf, Tw, qr, q, U, At, A, h, D, Cv, x, Hr, E, R, k, k0, 27constants: , Cp, U, D, Cv, Hr, E, R, k0 9
specifications: t 1driving forces: Fe, Tf, CAf 3
unknown variables: Fs, V, CA, rA, T, Tw, qr, q, A, At, h, x, k, 14
equations: 14
Degree of Freedom = variables – constants – specifications – driving forces – equations = unknown variables – equations = 27 – 9 – 1 – 3 – 14 = 0
Dynamic Degree of Freedom (index < 2) = differential equations = 3
Needs 3 initial condition: h(0), CA(0), T(0) 3
Consistency analysis
Multiplicity of Steady StatesMultiplicity of Steady States
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Running EMSO
Open MSO file
Multiplicity of Steady StatesMultiplicity of Steady States
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Consistency Analysis
Results
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The CSTR example at the steady state satisfy:
0
0
( )1( ) ( )
1
E
RTr Aft
f w EP RT
P
H k e CU AT T T T
V CC k e
e
V
F
01
AfA E
RT
CC
k e
Multiplicity of Steady StatesMultiplicity of Steady States
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Rewriting the energy balance:
( ) ( )R GQ T Q T0
0
( )( )
1
E
RTr Af
G E
RTP
H k e CQ T
C k e
Q T a T bR ( )
aU A
V Ct
P
1
f t w
P
T U A Tb
V C
T
Q
QG
QR3QR2QR1
1 2
3
4
5
GR dQdQ
dT dT
stable:
GR dQdQ
dT dT
unstable:
Multiplicity of Steady StatesMultiplicity of Steady States
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( , )dx
F t xdt ( ) 0F x
1( 1) ( ) ( ) ( )( ) ( ) , 0,1,2,k k m kx x J x F x k Newton-Raphson:
( )( ) ( )
( )k
k iij
j
F xJ x
x
and 0 1m k
Path FollowingPath Following
Homotopic Continuation: ( ; ) (1 ) ( ) ( ) 0 , 0 1H x p p F x pG x p
(0) (0)( ) ( ) ( )G x J x x x (0)( ) ( ) ( )G x F x F x
affine homotopy
Newton homotopy
Multiples solutions can be obtained by continuously varying the parameter p
Multiplicity of Steady StatesMultiplicity of Steady States
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Parametric Continuation: [ ( ); ( )] 0F x s p s where s is some parameterization, e.g., path arc length
( ) ( ) 0 , eF F dx dp
x s p s x px p ds ds
F FDF
x p
Frechet derivative
a point (xo, po) is:
( , )o oF x p
x
- Regular if is non-singular
( , )o oF x p
x
- Turning point if is singular and DF has rank = n
( , )o oF x p
x
- Bifurcation if is singular and DF has rank < n
reparameterization
Path FollowingPath FollowingMultiplicity of Steady StatesMultiplicity of Steady States
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Solutions: 1) CA = 13,13 kmol/m3 and T = 659,46 K 2) CA = 132,87 kmol/m3 and T = 523,01 K 3) CA = 299,86 kmol/m3 and T = 332,72 K
Example: a) execute flowsheet in file CSTR_noniso.mso with initial condition of 578 K and compare with result changing the initial condition to 579 K; b) find the three steady states using file CSTR_sea.mso by changing the initial guess for T and CA (use the section GUESS).
Multiplicity of Steady StatesMultiplicity of Steady States
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LinearizationLinearization
Generate linearized model at given operating point.
Implicit DAE:
Considering the specification as input, u(t), (SPECIFY section in EMSO):
And identifying the algebraic variables as y(t):
( ,́ , , , ) 0F x x y u t
( ,́ , ) 0F x x t
ˆ ˆ( ,́ , , ) 0F x x u t
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Differentiating F:
and extracting:
The partition:
Define the linearized system:
Linearization
´ 0x x y uF dx F dx F dy F du
1
x y x u
dx dxF F F F
dy du
1
x y x u
A BF F F F
C D
'x A x Bu
y C x Du
(index < 2)
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Example: execute the flowsheet in file CSTR_linearize.mso with the option Linearize = true and evaluate the characteristic values of the Jacobian matrix (matrix A). Repeat the example with the value of Cp 10 times smaller, i.e., 0.4 kJ / (kg K). Compare the ratio between the greater and the smaller characteristic values in module.
Non-isothermal CSTR: linearizationNon-isothermal CSTR: linearization
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Stability AnalysisStability Analysis
)(tx
( )dx
F xdt 0 0( ) ( )x t y t ( ) ( )x t y t
Liapunov Stability: is stable (or Liapunov stable) if, given > 0, there exists a = () > 0, such that, for any other solution, y(t), of
satisfying , then for t > t0.
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Stability AnalysisStability Analysis
)(tx
0 0( ) ( )x t y t b Asymptotic Stability: is asymptotic stable if Liapunov stable and there exists a constant b > 0 such that, if then 0)()(lim
tytx
t
( ) ( ) ( )y t x t x t Defining deviation variables:
2[ ( )]( ) ( ( ))
dx F x tF x F x t y O y
dt x
( ( ) )dx dx dy
F x t ydt dt dt
Expanding in Taylor series:
Linearization: [ ( )] ( )dy
J x t y A t ydt
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( )x tFor an equilibrium point = x*, the stability is characterized by the characteristics values of the Jacobian matrix J(x*) = A:
x* is a hyperbolic point if none characteristics values of J(x*) has zero real part.
x* is a center if the characteristics values are pure imaginary. Fixed point non-hyperbolic.
x* is a saddle point, unstable, if some characteristics values have real part > 0 and the remaining have real part < 0.
x* is stable or attractor or sink point if all characteristics values have real part < 0.
x* is unstable or repulsive or source point if at least one characteristic value have real part > 0.
Stability AnalysisStability Analysis
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For a second-order linear system:2det( ) ( ) det( ) 0A I tr A A
2( ) ( ) 4det( )
2
tr A tr A A
Stability AnalysisStability Analysis
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0 0( ) , (0)E
RTeAA A A A Af
FdCC C k e C C C
dt V
00
( ) ( )( ) , (0)
E
RTe r A t w
fp p
F H k e C UA T TdTT T T T
dt V C V C
Considering the CSTR example with constant volume:
0 02
0 02
( , )( ) ( )
E E
RT RTeA
E EART RT
r e r A t
p p p
F Ek e k e C
V RTJ C T
H k e F H k e C UAE
C V RT C V C
Stability AnalysisStability Analysis
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3 4
3 4
1.6458 10 3.4282 10(13.13, 659.46)
2.7542 10 4.8934 10J
4 4
4 4
1.6260 10 3.1753 10(132.87, 523.01)
1.5852 10 4.4509 10J
5 7
8 4
7.2050 10 6.6220 10(299.86, 332.72)
5.9285 10 1.0944 10J
5
4
7.2051 10
1.0944 10
5
4
6.3659 10
3.4614 10
3
4
1.0205 10
1.3604 10
2) Saddle Point, unstable
1) Stable node
3) Stable Node
Stability AnalysisStability Analysis
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300 50 100 150 200 250 300 350
300
350
400
450
500
550
600
650
700
750
800
CA
T
file: CSTR_nla/traj_cstr.m
Stability AnalysisStability Analysis
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Complex Dynamic BehaviorComplex Dynamic Behavior
Tw
Tw
CA
THopf point
Tw = 200,37 K
unstable solutions
stable solutions
CSTR example:
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t (h)
t (h)
unstable limit cycle
A limit cycle is stable if all characteristics values of exp(J p) (Floquet multipliers) are inside the unitary cycle, where J is the Jacobian matrix in the cycle, p = 2 / is the oscillation period and = |Hopf|.
file: CSTR_auto/cstr_bif.mso
Complex Dynamic BehaviorComplex Dynamic Behavior
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Interface EMSO-AUTOInterface EMSO-AUTO
parameters
Equation system
Jacobian matrix
First steady-state solution
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Interface EMSO-AUTOInterface EMSO-AUTO
21 ( )1 11 x tdx t
x t p x t edt
222 13 14 1 x tdx t
x t p x t edt
2 2
2 2
1
1
1 (1 )( )
14 3 14 (1 )
x x
x x
p e p x eJ x
p e p x e
p = 0: x* = (0, 0) (J) = (-1, -3)
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Interface EMSO-AUTOInterface EMSO-AUTO
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
1
2
3
4
5
6
7
8
9
10
x1
x 2
0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 0.20
0.1
0.2
0.3
0.4
0.5
0.6
0.7
0.8
0.9
1
x1
x 2
0.134 0.136 0.138 0.14 0.142 0.144 0.146 0.148
0.63
0.64
0.65
0.66
0.67
0.68
0.69
x1
x 2
Parameter p Eigenvalues Phase plane
p < 0.06361 Real negatives eigenvalues – stable nodep = 0.05 = [-1.13, -2.06]
p = 0.06361 Repeated real negatives eigenvalues – stable node (star) = [-1.4372, -1.4372]
0.06361 < p < 0.0889 Complex eigenvalues with negative real part - stable focusp = 0.085 = -1.095 0.565 i
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Interface EMSO-AUTOInterface EMSO-AUTO
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
1
2
3
4
5
6
7
8
9
10
x1
x 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
x1
x 2
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
x1
x 2
p = 0,0889unstable node gives rise to two points: unstable node and saddle point p > 0.0889
Turning point (fold): One stable solution (focus) and other unstable (node). (point 3 in figure below) = -1.009 0.605 i = [0, 3.432]
0.0889 < p < 0.0933 One stable solution (focus) and two unstable (saddle and node)p = 0.09 = -0.982 0.614 i = [-0.213, 3.332] = [0.364, 3.151]
0.0933 < p < 0.10574at p = 0.105738931 the first point goes from stable focus to stable node: = [-0.055, -0.046]
One stable solution (focus) and two unstable (saddle and focus)p = 0.10 = -0.652 0.651 i = [-0.439, 1.953] = 1.431 1.851 i
![Page 35: CSTR: Multiplicidade de Soluções e Análise de Estabilidade Argimiro R. Secchi Programa de Engenharia Química – COPPE/UFRJ Rio de Janeiro, RJ EQE038 – Simulação](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d355503460f94a0bedf/html5/thumbnails/35.jpg)
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Interface EMSO-AUTOInterface EMSO-AUTO
0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10
1
2
3
4
5
6
7
x1
x 2
0 2 4 6 8 10 12 14 16 18 200
2
4
6
8
10
12
t
x
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 1.8 20
2
4
6
8
10
12
x1
x 2
0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 13
3.5
4
4.5
5
5.5
6
x1
x 2
p = 0.10574the stable node gives rise to two points: stable node and saddle forp < 0.10574
Turning point (fold): One stable solution (node), other unstable (focus), and one sable limit cycle. (point 2 in figure below) = [-0.097, 0] = 1.186 2.478 i
0.10574 < p < 0.1309 One unstable solution (focus) and one stable limit cyclep = 0.12 = 0.528 3.487 i
p = 0.1309 Hopf bifurcation: pure imaginary eigenvalues. (point 4 in figure below) = 4.008 i
![Page 36: CSTR: Multiplicidade de Soluções e Análise de Estabilidade Argimiro R. Secchi Programa de Engenharia Química – COPPE/UFRJ Rio de Janeiro, RJ EQE038 – Simulação](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d355503460f94a0bedf/html5/thumbnails/36.jpg)
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Interface EMSO-AUTOInterface EMSO-AUTO
0.8 0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98 13
3.5
4
4.5
5
5.5
6
x1
x 2
p > 0.1309 Complex eigenvalues with negative real part - stable focusp = 0.15 = -0.952 4.627 i
p
x 2
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Interface EMSO-AUTOInterface EMSO-AUTO
-4 -3 -2 -1 0 1 2 3 4-10
-8
-6
-4
-2
0
2
4
6
8
10
Re()
Im(
)
Hopf
Hopf
1st turning point
2nd turning point
Trajectories:stable pointsaddle pointunstable point
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Interface EMSO-AUTOInterface EMSO-AUTO
Example: copy files auto_emso.exe and r-emso.bat (Windows) or @r-emso (linux) in “bin” folder of EMSO to the folder CSTR_auto and execute the command below in a prompt of commands (shell):
Windows: r-emso cstr_bif
Linux: ./@r-emso cstr_bif
The results are stored in file fort.7. In Linux the graphic tool PLAUT can be used to plot the results using the command @p.
![Page 39: CSTR: Multiplicidade de Soluções e Análise de Estabilidade Argimiro R. Secchi Programa de Engenharia Química – COPPE/UFRJ Rio de Janeiro, RJ EQE038 – Simulação](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d355503460f94a0bedf/html5/thumbnails/39.jpg)
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Sensitivity AnalysisSensitivity Analysis
Objective: determine the effect of variation of parameters (p) or input variables (u) on the output variables.
Steady-state simulation: );,(
0);,(
puxHy
puxF
Sensitivity analysis
local:
global: bifurcation diagram, surface response
,
iy
j x p
yW
p
,
ix
j x p
xW
p
(case study)
1
x
F FW
x p
y x
H HW W
x p
,
j iy
i j x p
p yW
y p
Normalized form:
![Page 40: CSTR: Multiplicidade de Soluções e Análise de Estabilidade Argimiro R. Secchi Programa de Engenharia Química – COPPE/UFRJ Rio de Janeiro, RJ EQE038 – Simulação](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d355503460f94a0bedf/html5/thumbnails/40.jpg)
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Sensitivity AnalysisSensitivity Analysis
Dynamic simulation: 0 0( , , , ; ) 0 , ( ; ) ( )
( , ; )
F t x x u p x t p x p
y H x u p
00( ) ( ) 0 , ( )x x x
xF F FW t W t W t
x x p p
p
HtW
x
HW xy
)(
where ( ) xx
dWW t
dt( )x
xW t
p
![Page 41: CSTR: Multiplicidade de Soluções e Análise de Estabilidade Argimiro R. Secchi Programa de Engenharia Química – COPPE/UFRJ Rio de Janeiro, RJ EQE038 – Simulação](https://reader035.vdocuments.net/reader035/viewer/2022062421/56649d355503460f94a0bedf/html5/thumbnails/41.jpg)
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Sensitivity AnalysisSensitivity Analysis
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Sensitivity AnalysisSensitivity Analysis
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ReferencesReferences
DAE Solvers:
DASSL: Petzold, L.R. (1989), http://www.enq.ufrgs.br/enqlib/numeric/numeric.html
DASSLC: Secchi, A.R. (1992), http://www.enq.ufrgs.br/enqlib/numeric/numeric.html
MEBDFI: Abdulla, T.J. and J.R. Cash (1999), http://www.netlib.org/ode/mebdfi.f
PSIDE: Lioen, W.M., J.J.B. de Swart, and W.A. van der Veen (1997), http://www.cwi.nl/cwi/projects/PSIDE/
SUNDIALS: Serban, R. et al. (2004), http://www.llnl.gov/CASC/sundials/description/description.html